The real line wrapped around a circle

The principle of winding

Imagine a real-valued line segment, tangent to the unit circle at point I (0, 1) on the circle… or, in fact, at point I(1, 0). We ‘wrap’ this line around the circle: each real number t is thus associated with a unique point M on the circle, obtained by travelling a distance t along the circle from I (in the clockwise direction if t > 0, and anti-clockwise if t < 0).

Practical example

The real number t = π corresponds to half a turn of the circle (the length of the full circle is 2π, as the radius = 1). We therefore arrive at point I’(-1, 0). The real number t = π/2 corresponds to a quarter of a turn: we arrive at J(0, 1).

One point, several real numbers

As the circle has a circumference of 2π, a real number t and the real number t + 2π (or t – 2π, or more generally t + k × 2π for k an integer) both give exactly the same point M on the circle. We say that these real numbers are congruent modulo 2π.

Example

The real numbers 0, 2π, 4π and −2π all correspond to the same point I(1, 0).

Common pitfall

A point on the circle corresponds to an infinite number of real numbers (all congruent modulo 2π), but a given real number corresponds to a single point. Do not reverse this direction: ‘a real number → a single point’ is true; the converse is false.